How does one clean a soiled tallit? It is made of wool and is quite old, and has never been cleaned.
A wool tallit needs to be hand-washed with a delicate soap, like Woolite, which is especially designed for delicate, wool garments. Soak it for several hours or overnight and then hand wash. If necessary, repeat more than once.
A tallit can also be dry cleaned. However, it might not be advisable to give a tallit to your local dry leaning establishment—as the ones not familiar with tallitot (i.e. most cleaners that are not situated in areas with orthodox Jewish populations) don't always know how to handle them and can cause damage.